Getting a result like $0<1$ means that the claim is true for all x. In other words, for any $x$ you input, the inequation holds true. (For all of your steps taken were equivalent to their predecessor, and as such did not change the truth/false value of the inequation. In some cases, mathematicians like to denote that by adding "$\Leftrightarrow$" at the start of each line.)

And there is a error in your computation aswell. I know no (reasonable) definition of $\mathbb{Z}$ such that $10-6=-4$

You are correct. The conclusion you came to means that for all $x$ the inequality you stated is true

EDIT: As De Vito pointed out, if you arrive correctly at a false statement this necessarily means that the assumption is false. In other words if the implication $P\Rightarrow Q$ is true and $Q$ is false then $P$ is also false.

But, if you arrive correctly at a true statement, this doesn't necessarily mean that the assumption is true. For that you will need $P\Leftrightarrow Q$ and not simply $P\Rightarrow Q$. In our case, you have shown $5(x + 2) + 2(x-3) < 3(x - 1) + 4x\Leftrightarrow 4<-3$